Balanced Bipartite Graph Research Articles

Minimum degree stability of C2k+1 ${C}_{2k+1}$‐free graphs

AbstractWe consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a ‐free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist showed that for , if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist also pointed out that this result cannot be extended to . In this paper, we give a complete answer for any . We show that if and is an ‐vertex ‐free graph with , then is bipartite, and the bound is tight.

“Less” Strong Chromatic Indices and the (7, 4)-Conjecture

A proper edge coloring of a graph 𝐺 is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings). The strong chromatic index 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞𝐴(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring. The second is 𝑞𝐵(𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a B-coloring. These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞𝐴(𝐺), 𝑞𝐵(𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special(7, 4)-conjecture. In order to understand the behavior of 𝑞𝐴(𝐺) and 𝑞𝐵(𝐺), we study these parameters for some graphs.We note that 𝑞𝐴(𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞𝐵(𝐺) is studied here for the first time.

Hamilton cycles passing through a matching in a bipartite graph with high degree sum

Let G be a balanced bipartite graph of order 2n with partite sets X and Y, and M be a matching of k edges in G. Let σ1,1(G)=min⁡{dG(x)+dG(y):x∈X,y∈Y,xy∉E(G)}. Conditions on σ1,1(G) for the existence of a Hamilton cycle passing through M have been studied by many researchers, but the threshold is not known to be sharp for 2n3≤k≤n−2.In this paper we prove that G has a Hamilton cycle passing through all the edges of M if σ1,1(G)≥2n−k and k≤n−2. Moreover, we show that the lower bound on σ1,1(G) is sharp for every k with 2n3≤k≤n−2.

Spectral extremal graphs for the bowtie

Let Fk be the (friendship) graph obtained from k triangles by sharing a common vertex. The Fk-free graphs of order n which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that n is sufficiently large. In this paper, we get rid of the condition on n being sufficiently large if k=2. The graph F2 is also known as the bowtie. We show that the unique n-vertex F2-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if n≥7, and the condition n≥7 is tight. Our result is a spectral generalization of a theorem of Erdős, Füredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that ex(n,F2)=⌊n2/4⌋+1. Moreover, we study the spectral extremal problem for Fk-free graphs with given number of edges. In particular, we show that the unique m-edge F2-free spectral extremal graph is the join of K2 with an independent set of m−12 vertices if m≥8, and the condition m≥8 is tight.

Group connectivity of graphs satisfying the Chvátal-condition

Let G be a (simple) graph on n≥3 vertices and (d1,…,dn) be the degree sequence of G with d1≤⋯≤dn. The classical Chvátal’s theorem states that if dm≥m+1 or dn−m≥n−m for each m with 1≤m<n2 (called the Chvátal-condition), then G is hamiltonian. Similarly, let G be a (simple) balanced bipartite graph on n≥4 vertices and (d1,…,dn) be the degree sequence of G with d1≤⋯≤dn. The classical Chvátal’s theorem states that if dm≥m+1 or dn2≥n2−m+1 for each m with 1≤m≤n4 (called the Chvátal-condition), then G is hamiltonian. In this paper, for an abelian group A of order at least 4, we show that if a graph G satisfies the Chvátal-condition, then G is A-connected if and only if G≠C4, where Cℓ is a cycle of length ℓ. Moreover, for an abelian group A of order at least 5, we also show that if a balanced bipartite graph G satisfies the Chvátal-condition, then G is A-connected if and only if G≠C6.

Extremal bipartite independence number and balanced coloring

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree D and with n vertices on each side has a balanced independent set containing (1−ϵ)logDDn vertices from each side for small ϵ>0. Secondly, we prove that the vertex set of every sufficiently large balanced bipartite graph with maximum degree at most Δ can be partitioned into (1+ϵ)ΔlogΔ balanced independent sets. Both of these results are algorithmic and best possible up to a factor of 2, which might be hard to improve as evidenced by the phenomenon known as ‘algorithmic barrier’ in the literature. The first result improves a recent theorem of Axenovich, Sereni, Snyder, and Weber in a slightly more general setting. The second result improves a theorem of Feige and Kogan about coloring balanced bipartite graphs.

Unavoidable patterns in locally balanced colourings

AbstractWhich patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? We show that when $\varepsilon \gt 1/4$ , $K_n$ must contain a complete subgraph on $\Omega (\log n)$ vertices where one of the colours forms a balanced complete bipartite graph.When $\varepsilon \leq 1/4$ , this statement is no longer true, as evidenced by the following colouring $\chi$ of $K_n$ . Divide the vertex set into $4$ parts nearly equal in size as $V_1,V_2,V_3, V_4$ , and let the blue colour class consist of the edges between $(V_1,V_2)$ , $(V_2,V_3)$ , $(V_3,V_4)$ , and the edges contained inside $V_2$ and inside $V_3$ . Surprisingly, we find that this obstruction is unique in the following sense. Any two-colouring of $K_n$ in which each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours (with $\varepsilon \gt 0$ ) contains a vertex set $S$ of order $\Omega _{\varepsilon }(\log n)$ on which one colour class forms a balanced complete bipartite graph, or which has the same colouring as $\chi$ .

Degree sum condition on distance 2 vertices for hamiltonian cycles in balanced bipartite graphs

Let G be a 2-connected balanced bipartite graph of order 2n with n≥3. Denote μ(G)=min⁡{max⁡{d(x),d(y)}:x,y∈V(G),dist(x,y)=2} and μ2(G)=min⁡{d(x)+d(y):x,y∈V(G),dist(x,y)=2}. Wang and Liu (2018) [14] proved that if μ(G)≥n+12, then G is hamiltonian. In this paper, we characterize non-hamiltonian bipartite graphs with μ(G)=n2. Also, we show that G is bipancyclic with one exception, if μ(G)≥n+12. As a direct consequence, we also show that if μ2(G)≥n+1, then G is hamiltonian and, with one exception, G is bipancyclic. Moreover, if we replace n+1 with n+2, then G contains a hamiltonian cycle passing through every edge of a perfect matching. The lower bounds in all results are sharp.

On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G) = RT(G) ? RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RDL(G) is a real symmetric matrix, we denote its eigenvalues as ?1(RDL(G)) ? ?2(RDL(G)) ?...? ?n(RDL(G)). The largest eigenvalue ?1(RDL(G)) of RDL(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RDL(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph Sn is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph Kn, Kn ? e, the star Sn, the complete balanced bipartite graph Kn/2 ,n/2 and the complete split graph CS(n, ?) are all determined from the RDL-spectrum.

Disjoint cycles covering specified vertices in bipartite graphs with partial degrees

Let G be a balanced bipartite graph of order 2n with bipartition (X,Y), and S a subset of X. Suppose that every pair of nonadjacent vertices {x,y} with x∈S,y∈Y satisfies dG(x)+dG(y)≥n+1. We show that if |S|≥2k+1, then G contains k disjoint cycles such that each of them contains at least two vertices of S. Moreover, if |S|≥2k+2, then G contains k disjoint cycles covering S such that each of them contains at least two vertices of S. Here, the lower bounds of |S| are necessary, and for the latter result the degree condition is sharp.

Ramsey properties of algebraic graphs and hypergraphs

Abstract One of the central questions in Ramsey theory asks how small the largest clique and independent set in a graph on N vertices can be. By the celebrated result of Erdős from 1947, a random graph on N vertices with edge probability $1/2$ contains no clique or independent set larger than $2\log _2 N$ , with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools. Say that an r-uniform hypergraph $\mathcal {H}$ is algebraic of complexity $(n,d,m)$ if the vertices of $\mathcal {H}$ are elements of $\mathbb {F}^{n}$ for some field $\mathbb {F}$ , and there exist m polynomials $f_1,\dots ,f_m:(\mathbb {F}^{n})^{r}\rightarrow \mathbb {F}$ of degree at most d such that the edges of $\mathcal {H}$ are determined by the zero-patterns of $f_1,\dots ,f_m$ . The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity $(n,d,m)$ has good Ramsey properties, then at least one of the parameters $n,d,m$ must be large. In 2001, Rónyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if G is an algebraic graph of complexity $(n,d,m)$ on N vertices, then either G or its complement contains a complete balanced bipartite graph of size $\Omega _{n,d,m}(N^{1/(n+1)})$ . We extend this result by showing that such G contains either a clique or an independent set of size $N^{\Omega (1/ndm)}$ and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for r-uniform algebraic hypergraphs that are defined by a single polynomial that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.

Transversals and Bipancyclicity in Bipartite Graph Families

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geqslant 4$, every balanced bipartite graph on $2n$ vertices in which each vertex in one color class has degree greater than $\frac{n}{2}$ and each vertex in the other color class has degree at least $\frac{n}{2}$ is bipancyclic. We prove a generalization of this theorem in the setting of graph transversals. Namely, we show that given a family $\mathcal{G}$ of $2n$ bipartite graphs on a common set $X$ of $2n$ vertices with a common balanced bipartition, if each graph of $\mathcal G$ has minimum degree greater than $\frac{n}{2}$ in one color class and minimum degree at least $\frac{n}{2}$ in the other color class, then there exists a cycle on $X$ of each even length $4 \leqslant \ell \leqslant 2n$ that uses at most one edge from each graph of $\mathcal G$. We also show that given a family $\mathcal G$ of $n$ bipartite graphs on a common set $X$ of $2n$ vertices meeting the same degree conditions, there exists a perfect matching on $X$ that uses exactly one edge from each graph of $\mathcal G$.

The spanning k-trees, perfect matchings and spectral radius of graphs

A k-tree is a spanning tree in which every vertex has degree at most k. In this paper, we provide a sufficient condition for the existence of a k-tree in a connected graph with fixed order in terms of the adjacency spectral radius and the signless Laplacian spectral radius, respectively. Also, we give a similar condition for the existence of a perfect matching in a balanced bipartite graph with fixed order and minimum degree.

Effective resistances and spanning trees in the complete bipartite graph plus a matching

In 2019, Ye and Yan computed the effective resistances in the nearly balanced complete bipartite graph Kn,n−pK2(p≤n). Then the result was extended to Km,n−pK2(p≤min{m,n}) very recently. In this paper, we obtain the effective resistances and the number of spanning trees in any complete bipartite graph plus a matching.

Perfect matching and distance spectral radius in graphs and bipartite graphs

A perfect matching in a graph G is a set of nonadjacent edges covering every vertex of G. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a distance spectral radius condition to guarantee the existence of a perfect matching. Let G be an n-vertex connected graph where n is even and λ1(D(G)) be the distance spectral radius of G. Then the following statements are true.(I) If 4≤n≤8 and λ1(DG)≤λ1(D(Sn,n2−1)), then G contains a perfect matching unless G≅Sn,n2−1 where Sn,n2−1≅Kn2−1∨(n2+1)K1.(II) If n≥10 and λ1(DG)≤λ1(D(G∗)), then G contains a perfect matching unless G≅G∗ where G∗≅K1∨(Kn−3∪2K1).Moreover, if G is a connected 2n-vertex balanced bipartite graph with λ1(D(G))≤λ1(D(Bn−1,n−2)), then G contains a perfect matching, unless G≅Bn−1,n−2 where Bn−1,n−2 is obtained from Kn,n−2 by attaching two pendent vertices to a vertex in the n-vertex part.

Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.

Extremal problems on distance spectra of graphs

Motivated by recent progresses on spectral extremal graph theory, in this paper, we aim to investigate the existence of cycles with given length in a graph in terms of its distance spectral radius. First of all, we show that if G is a connected bipartite graph with λ1(D(G))≤λ1(D(K1,n−1)), then G contains a C4 unless G≅K1,n−1. When n is sufficiently large with respect to k, as a corollary, we show that Sk(D(G))≥2n−2k if G is a C4-free bipartite graph. Besides, we prove that Sk(D(G))≥2n−2k if G is a bipartite distance regular graph. These two results partially solve a problem proposed by Lin (2019). Secondly, we give sufficient conditions for the existence of a Hamilton cycle or Hamilton path in a balanced bipartite graph in terms of the distance spectral radius.

Bipartite Independent Number and Hamilton-Biconnectedness of Bipartite Graphs

Let G be a balanced bipartite graph with bipartite sets X, Y. We say that G is Hamilton-biconnected if there is a Hamilton path connecting any vertex in X and any vertex in Y. We define the bipartite independent number $$\alpha ^o_B(G)$$ to be the maximum integer $$\alpha $$ such that for any integer partition $$\alpha =s+t$$ , G has an independent set formed by s vertices in X and t vertices in Y. In this paper we show that if $$\alpha ^o_B(G)\le \delta (G)$$ then G is Hamilton-biconnected, unless G has a special construction.

An approximate version of Jackson’s conjecture

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for everyε &gt; 0 there existsn0such that every diregular bipartite tournament on 2n≥n0vertices contains a collection of (1/2–ε)ncycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for everyc &gt; 1/2 andε &gt; 0 there existsn0such that everycn-regular bipartite digraph on 2n≥n0vertices contains (1−ε)cnedge-disjoint Hamilton cycles.

Some generalizations of spectral conditions for 2s-hamiltonicity and 2s-traceability of bipartite graphs

ABSTRACT This paper mainly focuses on spectral conditions for the Hamiltonicity of balanced bipartite graphs and nearly balanced bipartite graphs. Let and be the spectral radius and signless Laplacian spectral radius of a graph G, respectively. One of our main results is the following theorem: Let G be a balanced bipartite graph of order and of minimum degree . If and , then G is -hamiltonian unless . If and , then G is -hamiltonian unless . This theorem generalizes the results in [B. L. Li, B. Ning, Spectral analogues of Erdos' and Moon–Moser's theorems on Hamilton cycles, Linear and Multilinear Algebra, 64 (2016) 2252–2269].